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2012

Volume 10, Issue 2 (partial)


Effective Pressure Interface Law for Transport Phenomena between an Unconfined Fluid and a Porous Medium Using Homogenization

Anna Marciniak-Czochra and Andro Mikelić

Multiscale Model. Simul. 10, pp. 285-305 (21 pages)

Online Publication Date: April 03, 2012

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We present modeling of the incompressible viscous flows in the domain containing unconfined fluid and a porous medium in the case when the flow in the unconfined domain dominates. For such a setting a rigorous derivation of the Beavers–Joseph–Saffman interface condition was undertaken by Jäger and Mikelić [SIAM J. Appl. Math., 60 (2000), pp. 1111–1127] using the homogenization method. So far the interface law for the pressure was conceived and confirmed only numerically. In this article we derive the Beavers and Joseph law for a general body force by estimating the pressure field approximation. Different from the Poiseuille flow case, the velocity approximation is not divergence-free and the precise pressure estimation is essential. This new estimate allows us to rigorously justify the pressure jump condition using the Navier boundary layer, already used to calculate the constant in the law by Beavers and Joseph. Finally, our results confirm that the position of the interface influences the solution only at the order of physical permeability and therefore the choice of this position does not pose problems.

Dark Solitons, Dispersive Shock Waves, and Transverse Instabilities

M. A. Hoefer and B. Ilan

Multiscale Model. Simul. 10, pp. 306-341 (36 pages)

Online Publication Date: April 12, 2012

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The nature of transverse instabilities of dark solitons for the (2+1)-dimensional defocusing nonlinear Schrödinger/Gross–Pitaevskiĭ (NLS/GP) equation is considered. Special attention is given to the small (shallow) amplitude regime, which limits to the Kadomtsev–Petviashvili (KP) equation. We study analytically and numerically the eigenvalues of the linearized NLS/GP equation. The dispersion relation for shallow solitons is obtained asymptotically beyond the KP limit. This yields (1) the maximal growth rate and associated wavenumber of unstable perturbations and (2) the separatrix between convective and absolute instabilities. The instability properties of the dark soliton are directly related to those of oblique dispersive shock wave (DSW) solutions. Stationary and nonstationary oblique DSWs are constructed analytically and investigated numerically by direct simulations of the NLS/GP equation. It is found that stationary and nonstationary oblique DSWs have the same jump conditions in the shallow and hypersonic regimes. These results have application to controlling nonlinear waves in dispersive media.

Multiscale Developments of the Cellular Potts Model

M. Scianna and L. Preziosi

Multiscale Model. Simul. 10, pp. 342-382 (41 pages)

Online Publication Date: April 12, 2012

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Multiscale problems are ubiquitous and fundamental in all biological phenomena that emerge naturally from the complex interaction of processes which occur at various levels. A number of both discrete and continuous mathematical models and methods have been developed to address such an intricate network of organization. One of the most suitable individual cell-based model for this purpose is the well-known cellular Potts model (CPM). The CPM is a discrete, lattice-based, flexible technique that is able to accurately identify and describe the phenomenological mechanisms which are responsible for innumerable biological (and nonbiological) phenomena. In this work, we first give a brief overview of its biophysical basis and discuss its main limitations. We then propose some innovative extensions, focusing on ways of integrating the basic mesoscopic CPM with accurate continuous models of microscopic dynamics of individuals. The aim is to create a multiscale hybrid framework that is able to deal with the typical multilevel organization of biological development, where the behavior of the simulated individuals is realistically driven by their internal state. Our CPM extensions are then tested with sample applications that show a qualitative and quantitative agreement with experimental data. Finally, we conclude by discussing further possible developments of the method.

Bose–Einstein Condensation beyond Mean Field: Many-Body Bound State of Periodic Microstructure

Dionisios Margetis

Multiscale Model. Simul. 10, pp. 383-417 (35 pages)

Online Publication Date: April 24, 2012

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We study stationary quantum fluctuations around a mean field limit in trapped, dilute atomic gases of repulsively interacting bosons at zero temperature. Our goal is to describe quantum-mechanically the lowest macroscopic many-body bound state consistent with a microscopic Hamiltonian that accounts for inhomogeneous particle scattering processes. In the mean field limit, the wave function of the condensate (macroscopic quantum state) satisfies a defocusing cubic nonlinear Schrödinger-type equation, the Gross–Pitaevskii equation. We include consequences of pair excitation, i.e., the scattering of particles in pairs from the condensate to other states, proposed in [T. T. Wu, J. Math. Phys., 2 (1961), pp. 105–123]. Our derivations rely on an uncontrolled yet physically motivated assumption for the many-body wave function. By relaxing mathematical rigor, from a particle Hamiltonian with a spatially varying interaction strength we derive via heuristics an integro–partial differential equation for the pair collision kernel, $K$, under a stationary condensate wave function, $\Phi$. For a scattering length with periodic microstructure of subscale $\epsilon$, we formally describe via classical homogenization the lowest many-body bound state in terms of $\Phi$ and $K$ up to second order in $\epsilon$. If the external potential is slowly varying, we solve the homogenized equations via boundary layer theory. As an application, we describe the partial depletion of the condensate.
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